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Inventory Theory

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Submitted By gashwini
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Megapop Inventory Model
Assumptions:
Lead time is constant
Multi-period model
Demand follows normal distribution (follow the histogram in excel sheet)
Model with lost sale and discounted quantities
Parameters:
Q: order quantity
Qo: optimal order quantity
D: Annual demand
H: holding cost per item per year
Co: order cost
C: cost per item n: number of orders per year r: order point ps : shortage cost p : lost profit per unit shortage
B(r) : expected shortage
L: lead time μ: average demand per week σ: standard deviation of demand μL: average demand in lead time σ: standard deviation of in lead time
We will optimize order policy for number of bushels. Hence we have converted weekly demand of popcorn into its corresponding weekly demand of number of bushels of corn as a raw material. * Observe histogram from excel attachment which clearly predict that demand of number of bushels of corn per week follows normal distribution with μ = 73.5 and σ = 13.8835 * Hence μL = μ * L = 73.5*3 = 220.5
And, σL = σ*L = 13.8835*3 = 24.0469 * H = $4*12/12 = $4 * L = 3 weeks * D = μ*52 = 73.5*52 = 3822 bushels per annum * Co = $(950 + 0.55Q)
i) C = 6.20 + 9% of 6.20 = 6.20 + 0.558 = $6.758 if Q ≤ 100 ii) If order is greater than 100 then 6% discount would be there.
6.20 – 6% of 6.20 = 6.20 – 0.375 = $5.828
C = 5.828 + 9% of 5.828 = 5.828 + 0.5245 = $6.3525 a) Q = 2DCOH = 2*3822*(950+0.55Q)4
Solving this equation for Q we get,
Q = 1971.7706
We have two interval for Q. (0, 100] and (100, ∞)
Since Q > 100 it falls under second interval and we get discount of 6% on each bushel.
Shortage cost per bag of popcorn = $0.16
Let find the shortage cost for shortage per bushel of corn,
1 bag of popcorn weights 200gm contributes 160 gm of corn.
1 bushel contributes 25401.5 gm of corn.
Hence, shortage cost per bushel of corn be ps
0.16160 = ps25401.5 ps = $25.40 for shortage per bushel

Each popcorn bag is sold for $1.6 with a 5% profit margin
Profit = $0.0762
1 bag of popcorn consumes 160gm of corn as a raw material.
That means 160 gm of corn contributes profit of $0.0762
Hence 1 bushel of corn (25401.5 gm) contributes profit p :
0.0762160 = p25401.5 p = $12.0975
Hence lost profit per unit shortage of corn bushel is $12.0975
For a given value of Q the reorder level r, is given by the smallest value of r having the cumulative lead time demand probability such that,
P(D ≥ r) ≥ ps+pDHQ+ps+pD
P(z ≥ r-μLσL) ≥ 25.40+12.0975*3822 4*1971.7706+ 25.40+12.0975*3822 = 0.9478
P(z ≥ r-220.524.0469) ≥ 0.9478
Corresponding z value is 1.62 * 1.62 ≥ r-220.524.0469
Solving this equation for r we get, r = 259.4559
Here, r > μL
B(r) = σL * L(r - μLσL) = 24.0469*L(1.62) (use Normal Loss Function Table) = 24.0469 * 0.022 = 0.529
Improved Q1 = 2D(CO+ pS + pB(r)H = 2*3822(950+0.55Q1 + 25.40 +12.0975*0.529)4 = 1984.817
‘r1’ for Q1 becomes 259.4559
B(r1) = 0.529
Repeating the same procedure again for Q2 and r2:
Q2 = 1984.817 r2 = 259.4559
B(r2) = 0.529 we get, Q1 = Q2 and r1 = r2
Hence optimal order quantity is 1984.817 bushels.
Number of orders: n = DQO = 38221984.817 = 1.9256
For further calculations we use optimal values : r = 259.4559 and B(r) = 0.529 b) We reorder when inventory reaches to 259.4559 bushels. c) Each inventory order cycle starts when ordered goods arrive.
Since number of orders are 1.9256 per year means we place order after every 6.23 months. Hence we can say that there are 2 order cycles per year on an average.
Probability running out of stock = P(D ≥ r)
P(D ≥ r) = P(z ≥ 259.4559-220.524.0469) = P(z ≥ 1.62) = 1 – P(z < 1.62) = 1 – 0.9474 = 0.0526
Hence there are 5.26% chances that Megapop runs out of stock. d) Total expected cost = annual order cost + expected annual holding cost + expected annual Shortage cost + purchase cost
Annual order cost = DHQO = 3822*(950+0.55*1984.817)1984.817 = $3931.4374
Expected annual holding cost = HQO2+ r- μL+ B(r) = 41984.8172+ 259.4559- 220.5+ 0.529 = $4127.5736
Expected annual Shortage cost = DQO(pS + p)B(r) = 38221984.817(25.40 + 12.0975)*0.529 = $38.1969
Purchase cost = C*D = 6.3525*3822 = $24279.255
(Note that purchase cost would remain same in every case since QO >100 so, both C and D are constants)
Total expected cost = $3931.4374 + $4127.5736 + $38.1969 + $24279.255 = $32,376.4602 e) In this case p remains same but ps changes. Shortage cost per bag of popcorn = $0.08 Let find the shortage cost for shortage per bushel of corn,
1 bag of popcorn weights 200gm contributes 160 gm of corn.
1 bushel contributes 25401.5 gm of corn.
Hence, shortage cost per bushel of corn be ps
0.08160 = ps25401.5 ps = $12.7007 for shortage per bushel
Q = 1971.7706
For a given value of Q the reorder level r, is given by the smallest value of r having the cumulative lead time demand probability such that,
P(D ≥ r) ≥ ps+pDHQ+ps+pD
P(z ≥ r-μLσL) ≥ 12.7007+12.0975*3822 4*1971.7706+ 12.7007+12.0975*3822 = 0.9232
P(z ≥ r-220.524.0469) ≥ 0.9232
Corresponding z value is 1.426 * 1.426 ≥ r-220.524.0469
Solving this equation for r we get, r = 254.7908
Here, r > μL
B(r) = σL * L(r - μLσL) = 24.0469*L(1.43) (use Normal Loss Function Table) = 24.0469 * 0.035 = 0.8416
Improved Q1 = 2D(CO+ pS + pB(r)H = 2*3822(950+0.55Q1 + 12.7007 +12.0975*0.8416)4 = 1985.4939
‘r1’ for Q1 becomes 254.6465
B(r1) = 0.8536
Repeating the same procedure again for Q2 and r2:
Q2 = 1985.6887 r2 = 254.6465
B(r2) = 0.8536
Repeating the same procedure again for Q2 and r2:
Q3 = 1985.6887 r3 = 254.6465
B(r3) = 0.8536 we get, Q2 = Q3 and r2 = r3
Hence optimal order quantity is 1985.6887 bushels.
Number of orders: n = DQO = 38221985.6887 = 1.9248
For further calculations we use optimal value: r = 254.6465 and B(r) = 0.8536
We reorder when inventory reaches to 254.6465 bushels. Since risk decreased reorder point is also decreased.
Since number of orders are 1.9248 per year means we place order after every 6.23 months. Hence order cycle remains same (i.e. 2) .
Probability running out of stock = P(D ≥ r)
P(D ≥ r) = P(z ≥ 254.6465-220.524.0469) = P(z ≥ 1.42) = 1 – P(z < 1.42) = 1 – 0.9222 = 0.0778 Probability of stock out has been increased to 0.0778 Total expected cost = annual order cost + expected annual holding cost + expected annual Shortage cost + purchase cost
Annual order cost = DHQO = 3822*(950+0.55*1985.6887)1985.6887 = $3930.6344
Expected annual holding cost = HQO2+ r- μL+ B(r) = 41985.68872+ 254.6465- 220.5+ 0.8536 = $8082.7552 Expected annual Shortage cost = DQO(pS + p)B(r) = 38221985.6887(12.7007 + 12.0975)*0.8536 = $40.7431 Purchase cost = $24279.255 Total expected cost = $3930.6364 + $8082.7552 + $40.7431 + $24279.255 = $36,333.3897
As we decrease risk factor overall cost and order size is increased while reorder level and probability of stock out has been increased. f) In order to overcome the problem of risk quantification forever Megapop should increase the order quantity and reorder level that would minimize stock out probability as well.

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